arXiv subjects
Publications and source records attributed to Shahar Nevo.
In this paper we introduce elementary and completely explicit formulas for the derivative of any order of any function of the type 1/p, where p is a polynomial with known zeros.
We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict minimum doesn't have to be decreasing to the left nor increasing to the right of the minimum, we present a function whose derivative is discontinuous at one point and has a strict minimum at this point (i.e. it oscillates only in one direction), we compare several definitions of inflection point, and we discuss a general version of the theorem on the derivative of inverse functions where continuity of the inverse function is assumed merely at one point.
This paper deals with (finite or infinite) sequences of arbitrary independent events in some probability space. We find sharp lower bounds for the probability of a union of such events when the sum of their probabilities is given. The results have parallel meanings in terms of infinite series.
We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $α>1$ and $k\ge2$ the condition $$\frac{|f^{(k)}|}{1+|f|^α}(z)\le C \qquad \mbox{ for all } z\in D$$ does not imply quasi-normality.
We show that the family ${\cal F}_k$ of all meromorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\ge C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. The proof relies mainly on the Zalcman-Pang rescaling method.
We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $α>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot \log_q x \cdot f^{(k)}(x)}{1+|f(x)|^α}\le 0 $$ holds.
We show that a family ${\cal F}$ of meromorphic functions in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f^{(j)}|^α}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F}$$ (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $α>1$ are real numbers) is quasi-normal. Furthermore, if all functions in ${\cal F}$ are holomorphic, the order of quasi-normality of ${\cal F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.
Let F be a family of functions meromorphic in a domain D. If {|f|/(1+|f|^3):f in F} is locally uniformly bounded away from zero, then F is normal.
In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let $\CF$ be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and let $h(\not\equiv0)$ be a holomorphic function on $D$, such that $h(z)$ has no common zeros with any $f\in\CF$. Assume also that the following two conditions hold for every $f\in\CF$:\linebreak %{enumerate} [(a)] (a) $f(z)=0\Longrightarrow f'(z)=h(z)$ and %[(b)] (b) $f'(z)=h(z)\Longrightarrow|f^{(k)}(z)|\le c$, where $c$ is a constant. Then $\CF$ is normal on $D$. %{enumerate} A geometrical approach is used to arrive at the result which significantly improves the previous results of the authors, \textit{A criterion of normality based on a single holomorphic function}, Acta Math. Sinica, English Series (1) \textbf{27} (2011), 141--154 and of Chang, Fang, and Zalcman, \textit{Normal families of holomorphic functions}, Illinois Math. J. (1) \textbf{48} (2004), 319--337. We also deal with two other similar criterions of normality. Our results are shown to be sharp.
We extend Riemann's rearrangement theorem on conditionally convergent series of real numbers to multiple instead of simple sums.
In 1986, S.Y. Li and H.Xie proved the following theorem:Let k>=2 and let F be a family of functions meromorphic in some domain D, all of whose zeros are of multiplicity at least k. Then F is normal if and only if the family F_k={f^(k)/(1+|f^k+1|):f in F} is locally uniformly bounded in D. Here we give, in the case k=2, a counterexample to show that if the condition on the multiplicities of the zeros is omitted, then the local uniform boundedness of F_2 does not imply even quasi-normality. In addition, we give a simpler proof for the Li-Xie Theorem that does not use Nevanlinna Theory which was used in the original proof.
We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not necessarily normality.
Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple exept finitely many and T(r,h)=o{T(r,f)} as r tends to infinity, then f'=h has infinitely many solutions (including poles).
For every natural number $T,$ we write $\Ln T$ as a series, generalizing the known series for $\Ln 2.$
We show that a family of meromorphic functions in the unit disk $\dk$ whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each $f$ meromorphic in $\dk$ we have $\inf_{z\in\dk} f^#(z)\le \frac{1}{2}$$ where $f^#$ denotes the spherical derivative of $f$.
We extend the fundamental normality test due to Carathéodory in the sense of shared functions.
In this paper we calculate the collection of limit functions obtained by applying an extension of Zalcman's Lemma, due to X. C. Pang, to the non-normal family $\left\{f(nz):n\in\mathbb{N}\right\}$ in $\mathbb{C}$, where $f=Re^P$. Here $R$ and $P$ are an arbitrary rational function and a polynomial, respectively, where $P$ is a non-constant polnomial.
For every countable ordinal number $α$ we construct an entire function $f=f_α$ such that the family $\left\{f(nz):n\in\mathbb{N}\right\}$ is exactly $Q_α$-normal in the unit disk.